Mitch Daniels was seen as a potential challenger to President Obama until he opted out of running last year. After his rebuttal of Obama's State of the Union address Tuesday, some GOP elites are openly longing for a Mitch Daniels candidacy.

Once upon a time, that was a key Republican Party strategy to win back the White House from President Obama. The two-term Indiana governor decided early last year to not seek the party's nomination, but the wishful thinking has only deepened after Mr. Daniels' concise and reasoned GOP rebuttal Tuesday night to Mr. Obama's State of the Union address.

“I could hear sighs all over the country from Republicans [about] what might have been,” conservative columnist Charles Krauthammer told Fox News late Tuesday. “That was one of the best speeches I’ve heard … and I think it was one of the best presentations of the conservative idea against the larger government of Obama.”

The first three GOP contests have not produced a definitive winner, and the Republican leadership, in picking Daniels to deliver the party's rebuttal on Tuesday, may be circling back to someone they perceive as able to “bridge the gap” between the far right faction of their party and the more doctrinaire establishment, says Bruce Buchanan, who teaches presidential politics and public policy at the University of Texas in Austin.

Daniels “strikes people clearly as a grownup … [and] the fact [that the Republican Party] invited Daniels and he accepted most likely indicates some hopes of attracting him,” Mr. Buchanan says.

Mr. Romney, who has a record as a moderate governor and whose Mormon faith is worrisome to evangelical Christian conservatives, and Mr. Gingrich, who resigned as speaker and whose marital infidelities are no secret, are perceived in some quarters as flawed. Compared with them, Daniels is largely free of controversy, and he has a record of working with both parties and of taking pragmatic positions on fiscal and social issues. He is also “more comfortable in his own skin” and “pretty unflappable,” which makes him less prone to gaffes than Romney or Gingrich, says Buchanan.

The national Republican Party courted Daniels for the 2012 race, but he bowed out last spring. His wife, Cheri Daniels, is reported to have vetoed the idea, citing privacy concerns. Mrs. Daniels and her husband divorced in 1993 and remarried four years later. In the interim, she married and divorced another man in California and was subsequently criticized for not sharing the responsibility of raising the four daughters with Daniels, a charge the governor publically said was false.

“They’re in a mess. They don’t want Gingrich and they don’t like Romney. They pushed Daniels to begin with. I don’t know if anything’s changed,” says Mr. Sabato.

At this point in the primary season, Daniels would need to be drafted as a nominee, a process that would involve the front-runner, either Romney or Gingrich, stepping down. That is an unlikely scenario.

Another potential roadblock is that Daniels, like Romney, will face accusations that he backtracks on issues, such as past statements he made about his disinterest in pursuing a so-called "right to work" law in Indiana. He now supports the legislation, designed to give workers the right to opt out of paying union dues – a move widely expected to undermine union organizing powers. It cleared the Indiana House late Wednesday.

Although Daniels is often described as a moderate for saying fiscal policy trumps social policy, his positions are “right in the mainstream of the conservative party,” says Marjorie Hershey, a political scientist at Indiana University in Bloomington.

Ms. Hershey doubts Daniels will change his mind about running for president. His wife’s concerns about family privacy are as just as salient now as they were last year, she says.

“A cabinet post might be an option for him or perhaps an ambassadorship.… However, I suspect he might be more at home on a corporate board and be done with all of this,” Hershey says.

Any discussion of Daniels’ future in this year’s presidential race is “hypothetical,” although it is certain “he’ll play a role in the conversation,” says Pete Seat, communications director for the Indiana Republican Party.

“The only reason that people are looking to him for guidance and for a vision is because of the results that we had in this state. We’ve taken those policies that the candidates are talking about and actually putting them into use,” Mr. Seat says.

One reason Daniels holds appeal for Republicans is an intangible one: He seems more authentic to voters than does the competition, says Seat. Daniels writes his own speeches, including Tuesday night’s rebuttal, and is known to be so close with the people in Indiana that he strikes up e-mail conversations with constituents and, when traveling, accepts invitations for overnight stays in their homes.

“He stays in the homes of Hoosiers when he travels around on the road. He’s sleeping in kids’ rooms with stuffed animals on the bed and Superman wallpaper," says Seat. "That’s how he keeps in contact with [his constituents]. He listens to people.”